---
tag: H1AT0G82ZD
kind: concept
name: Spread
msc: ["51E23", 51A40]
sources:
  - url: https://en.wikipedia.org/wiki/Spread_(projective_geometry)
same_as:
  wikidata: Q106611350
  wikipedia: Spread (projective geometry)
---

## Summary

A spread of a [[projective space|ZT9A6JK8QF]] is a family of subspaces of one fixed dimension that partitions the points.
Every point lies in exactly one member.
The best known case is a spread of lines of three-dimensional space.
Spreads matter because they build [[translation planes|JZ5J6REAMD]], through the André–Bruck–Bose construction.

## Explanations

### L1 human

Imagine filling all of space with lines so that no two lines meet and every point is on one line.
Such a family is a line spread.
Over the finite field with $q$ elements, three-dimensional projective space has $(q^2 + 1)(q + 1)$ points and each line has $q + 1$ of them, so a line spread has $q^2 + 1$ lines.
One such spread comes from viewing pairs of elements of the field with $q^2$ elements as a four-dimensional space over the field with $q$ elements.